3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA 327 have Di(/« p,) <; ( f/2"" d^Y" for all n 2> 1, and use Problem 12(e) of Section 13.11. Use this problem also to show that (y) implies (a).) (c) Let p ^> 1 and let p, be a positive measure on X. For each measure 9 on X, let dd/dp, denote the function h (defined up to equivalence) which features in the Lebesgue decomposition (13.18.4) 6 — h • ft + or, where |a| is disjoint from p.. Show that every representative measure for x has base p, if and only if for every positive measure v on X. (To show that the condition is necessary, reduce to the case p > 1 and Dp(v) > 0. Then apply Problem 10(d) and Holder's inequality.) (d) Let;? ^> 1 and let p be a positive measure on X with total mass 1. Then the follow- ing conditions are equivalent: (a) p, is the unique representative measure for x- (/3) Dp(v) = J^dv/dp,) for all positive measures v on X. (y) Dp(v) <^ J^dv/dp) for all positive measures v on X. (S) \\d6 ^ JM( | <#?/d]u, |) for all quasi-representative measures 0 for %. ("Szego- Kolmogoroff-Krein theorem"). (To prove that (y) implies (a), use (c) to show that p, is hyperextremal, and then use (b).) 12. With the hypotheses and notation of Problem 8, let /x be a representative measure for x- Let K.(p) denote the set of functions fe £>1(p) such that /• //, is a quasi- representative measure for x» in other words such that ufdp — ( /d/zll (u dp.\ - ft) - W|). We have B c K(JLI). for all u e B. Then (a) For each function g^.0 in -^(/x), show that there exists a function fe K(ft) such that |/| <;# and j fdp, = DI(|/| • p) = D^g - p) (use Problem 10(d)). (b) A function /e K(/x) is said to be exterior if \\fdp = Di(|/| • p) >0, and interior if |/(x)| = 1 almost everywhere with respect to p,. Let E(/x) (resp. I(/i)) be the set of exterior (resp. interior) functions. The intersection E(JU-) n I(/x) consists of functions equivalent to constants c such that |c| = 1. (c) Let g 1> 0 be a function in ^l(p) such that Dj(# • /x) « J^g) > 0. Show that there exists a function /e E(p) such that |/| ~<^. (Use (a) by writing/^^/z, where h e JS^Ou) and JM(|/r|) = 1, and use Problem 12(e) of Section 13.11.) 13. With the hypotheses and notation of Problem 8, let p, be a hyperextremal represen- tative measure for x- Then DI(/- p.) == J^(/) for all/ ^ 0 in &l(p) (Problem ll(b)). (a) Let/e K(/x). For/to be an exterior function (Problem 12), it is necessary and sufficient that inf I 11 — uf\ du — 0. (To show that this condition is sufficient, prove ueB J that it implies that there exists a sequence (wn) in B such that x(w/») ~ 1 fc>r all n an<^ . To show that the condition is necessary, we such that \unf\ dp, tends to \fd\t. • /x). Conclude that, for all functions /g> 0 in ^(ft), we).n particular, the image of C under the